All Questions
6,055 questions
0
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A Nomenclature Issue : Imprimitive Semigroup?
The following question was asked by me on the forum sci.math.research,
“An imprimitive group is a transitive permutation group with a non-trivial
equivalence relation compatible with the action of ...
- 38.6k
6
votes
1
answer
517
views
Growth zeta-functions of regular languages
Dear All,
my following question may be known and ought to be known, so in case it is folklore please could you give me the references.
To start, it is obvious that growth of rational languages are ...
- 1,951
12
votes
3
answers
3k
views
Can we say anything about the Krull dimension of a localization?
I'm looking for a theorem of the form
If $R$ is a nice ring and $v$ is a reasonable element in $R$ then Kr.Dim$(R[\frac{1}{v}])$ must be either Kr.Dim$(R)$ or Kr.Dim$(R)-1$.
My attempts to do ...
- 30.5k
7
votes
2
answers
2k
views
Global dimension and localization
Is there any condition on a commutative ring $R$ so that the global dimension of $R$ coincides with the supremum of the global dimensions of the localizations $R_{\mathfrak{m}}$ at all maximal ideals $...
- 30.3k
12
votes
2
answers
1k
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Cohen-Macaulay domain with non-Cohen-Macaulay normalization?
Is the normalization of a Cohen-Macaulay domain necessarily Cohen-Macaulay? I suspect that the answer is no, but I don't have a counterexample. I am most interested in "geometric" situations, so one ...
CommunityBot
- 1
29
votes
2
answers
5k
views
Regular, Gorenstein and Cohen-Macaulay
All the statements below are considered over local rings, so by regular, I mean a regular local ring and so on;
It is well-known that every regular ring is Gorenstein and every Gorenstein ring is ...
CommunityBot
- 1
2
votes
2
answers
1k
views
Irreducible component of a Cohen-Macaulay variety
Is it true that an irreducible component of a Cohen-Macaulay variety is also Cohen-Macaulay? If not, then in what cases does this fact hold?
- 20.5k
1
vote
1
answer
515
views
Cohen Macaulay, free and finitely generated module
Here is an unsolved problem for me in Kaplansky's "Commutative rings" p. 103, no. 18.
Let $R$ be a Cohen-Macaulay ring. Let $T$ be a ring containing $R$ and suppose that as an $R$-module it is free ...
- 42.9k
7
votes
1
answer
2k
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structure theorem for modules
Can structure theorem for modules be extended to modules over UFDS , to modules over Neotherian rings ? if yes then can one get the statement and reference?
Since operations on matrices with ...
- 25.8k
8
votes
1
answer
3k
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Number of graphs with a given number of nodes, edges and triangles
Hi. Does anyone know if it is possible to enumerate the set of labeled/unlabeled graphs (loopless, undirected, only one edge between pairs of nodes) having a given number of nodes, edges and triangles?...
- 902
3
votes
0
answers
916
views
Unibranch rings
Let us call a Noetherian local ring $A$ unibranch if it is a domain and the normalization map is finite and induces a bijection on spectra.
My question is as follows: is this property preserved when ...
- 1,209
7
votes
1
answer
433
views
Powers of maps on finite sets
Let $X_n$ be a set with $n$ elements. Write $F(X_n,X_n)$ for the set of maps from $X_n$ to itself. It is a monoid under the operation of composition. Let $m$ be a positive integer. How many maps in $...
- 13k
6
votes
3
answers
786
views
Trace of the identity map in a projective module
Let $A$ be a commutative algebra (over the complex numbers, with a unit) and let $M$ be a finitely generated projective $A$-module, and let $m_1,\ldots,m_n$ be a set of generators of $M$. The Dual ...
- 56.9k
4
votes
0
answers
811
views
$Ext$ functor, filtered complexes and spectral sequences
Let $\mathcal{A}$ an abelian category. Take $M$ an object of $\mathcal{A}$, and $K_*$ a bounded complex in $\mathcal{A}$ equipped with a bounded increasing filtration $F$. By using homological and ...
3
votes
0
answers
289
views
Terminal quasi-affine varieties?
Let $U$ be a normal, irreducible, quasi-affine variety over the algebraically
closed field $k$ and consider the ring $A = \mathscr{O}(U)$ of regular
functions on $U$. Write $Max(A)$ for the ...
- 42.9k
5
votes
0
answers
331
views
Extensions of maps between graded modules
Let $R$ be a connected graded ring (like $R=\mathbb R[x_1,\dots,x_d]$ with the usual grading) and let $N \subset R^{\oplus n}$ be a graded submodule, i.e. $$N= \bigoplus_{i \in \mathbb N} (N \cap R^{\...
- 25.5k
3
votes
1
answer
1k
views
Does totally ramified extension really exist?
The answer is certainly "Yes", but this is the problem I met in Algebraic Number Theory by Neukirch. I guess that I must be doing something wrong, since otherwise I will get the statement "There are ...
- 37k
5
votes
2
answers
367
views
Invariant means on commutative magmas
It is a very standard fact that commutative semigroups admit an invariant mean and the proof basically relies on Markov-Kakutani fixed point theorem. Now, it seems to me that the proof of this theorem ...
- 6,034
5
votes
0
answers
190
views
"Unknot" algebraic set defined by two mutually dependent set of variables
Let $n$ be an integer $\geq 4$, and let $V \subseteq {\mathbb C}^{2n-1}$ be the set of all
$(a_1,a_2, \ldots ,a_n,b_1,b_2, \ldots ,b_{n-1}) \in {\mathbb C}^{2n-1}$ such that the derivative of the ...
- 3,595
0
votes
0
answers
237
views
resolution of singular points on plane curves and base change
Let $k$ be a field and $C/k$ be an affine plane curve over $k$, namely $C = \mathrm{Spec}(A)$ for some $A = k[x,y]/(f(x,y))$, here $f(x,y) \in k[x,y]$ is an irreducible polynomial. Let $B$ be the ...
- 1,109
4
votes
1
answer
2k
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Primary decomposition and finitely generated abelian groups
In a question asked by Ben Webster, Harry Gindi commented that it is possible to prove the classification theorem from finitely generated abelian groups by appealing primary decomposition.
I have ...
CommunityBot
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15
votes
1
answer
4k
views
is residue field ever flat over its local ring?
Let R be a local ring with maximal ideal m and residue field k. Is k ever flat over R? What conditions are needed on R?
Sorry, it's not a very profound question. It came up in a derived functor ...
- 57.6k
0
votes
2
answers
617
views
An element in the product of schemes
Let $ f : X \to S, g : Y \to S$ be the scheme morphisms, and $ X \times_S Y$ be the product of shemes. Let $ z \in X \times_S Y$ , and $ x=p(z) , y=q(z) ,$ where $ p: X \times_S Y \to X, q: X \times_S ...
- 95
1
vote
1
answer
1k
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resolution of singular points on curve
After reading Fulton's book "Algebraic Curves", I know how to do resolution of singular points on curves. Given an affine equation, I can get it's non-singular affine model, i.e the normalization of ...
- 1,811
6
votes
2
answers
541
views
Positive matrices matrices over commutative rings
Assume that $R$ is a commutative ring with a ring compatible ordering and let $A$ and $B$ be symmetric $n\times n$ matrices with entries in $R$ such that $\sum x_iA_{ij}x_j\geq 0$ and $\sum x_iB_{ij}...
CommunityBot
- 1
13
votes
2
answers
1k
views
Is the Characteristic of a Field Detectable from the Topology of a Topological Vector Space?
Motivation
A topological vector space is a vector space over a (topological) field, K, that carries a topology such that addition and scalar multiplication are continuous maps, e.g., all normed vector ...
CommunityBot
- 1
2
votes
0
answers
1k
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Why is scalar extension important?
What I want to know is maybe not as dumb as the bare question.
Suppose B is a commutative unital ring and C is a category of B-modules. Suppose that f : A --> B is a homomorphism, and F is ...
- 403
1
vote
1
answer
325
views
Is a particular element of a particular ring a nonzerodivisor?
Let $A$ be the ring $\Bbbk[\alpha_0, \alpha_1, \alpha_2, x_0, x_1, x_2]$ (where $\Bbbk$ is an infinite field, algebraically closed if it matters). Let $g \in \Bbbk[\alpha_0, \alpha_1, \alpha_2]$ be a ...
- 5,857
7
votes
1
answer
912
views
Optimal reference for tensor product of symmetric bilinear forms?
This is just a reference request on a relatively elementary level (for which I apologize in advance), but every time I bump into this question I suspect I'm missing the "correct" conceptual setting. ...
- 66.3k
7
votes
4
answers
2k
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product of rings
I feel a need to apologies for this question, since it seems to be to basic to be asked.
in this question I am primarily concerned with commutative rings and therefore all rings here are assumed to ...
- 56.9k
5
votes
1
answer
540
views
Tensor product of regular ring (with some conditions)
Basically, my question is whether this answer is correct. Here is the point. Let $R$ be a ring, and let $A$ and $B$ be $R$-algebras. Suppose that $A$ is regular and $B \otimes_R B$ is regular too. ...
CommunityBot
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2
votes
1
answer
447
views
Commutator tensors and submodules
Let $k$ be a commutative ring with $1$, and let $B$ be a submodule of a $k$-module $A$.
For every $n\in\mathbb N$ and every $k$-module $V$, let $K^n\left(V\right)$ denote the kernel of the canonical ...
- 55.9k
2
votes
1
answer
281
views
Does fiberwise exactness imply exactness?
Let $R$ be a local Noetherian domain with fraction field $K$ and residue field $\Bbbk$. Let $C^{\bullet}$ be a bounded complex of free, finitely generated $R$-modules. Suppose that $C^{\bullet} \...
- 55.9k
4
votes
1
answer
552
views
Factorization of schemes
Let $k$ be a ring (perhaps a field). Let $M$ be the "set" of isomorphism classes of $k$-algebras and regard it as a commutative monoid with multiplication $\otimes_k $ and unit element $k$. There is ...
CommunityBot
- 1
3
votes
1
answer
538
views
Comparation of dimensions of rings
Let $k$ be a field, $A$ be an integral domain, $B \subset A$, and $A, B$ are both finitely generated $k$ algebra. Let $p \subset B$ be a prime ideal. Suppose there exists prime ideals $q \subset A$, ...
- 47.5k
13
votes
1
answer
1k
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Succesful applications of algebra in combinatorics
Hi. This may be a very general question.
Are there any examples of problems in combinatorics which were open, but which found a solution when stated in algebraic terms?
If yes, could somebody ...
- 2,955
1
vote
2
answers
292
views
Symbolic powers in regular local rings
I am having trouble understanding one of the results in the following paper
http://arxiv.org/PS_cache/math/pdf/0104/0104175v1.pdf
In proposition 3.1, the author says
Let $(R,\frak{m})$ be a ...
3
votes
0
answers
603
views
Norms in Galois extensions
Let $k$ be a field of characteristic 0, and $\overline k$ be a fixed algebraic closure of $k$.
Let $k\subset F\subset E$ be a tower of finite Galois extensions in $\overline k$,
where both $\mathrm{...
- 14.2k
4
votes
1
answer
1k
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checking if F[x]/I is isomorphic to F[x]/J
Let $F$ be a field. Let $p$ and $q$ be monic members $F[x]$. Let $I = \{p\cdot r : r\in F[x]\}$ and $J = \{q\cdot r : r\in F[x]\}$. I know that if $F[x]/I$ is isomorphic to $F[x]/J$ then ($\...
- 5,654
4
votes
1
answer
711
views
Faltings' category of almost modules
Hi,
Let $V$ be an integral domain with an ideal $m\subset V$ and put $K = S^{-1}V$ where
$S = {1}\cup m$ (a multiplicatively closed subset). Is it true that the category of almost
$(V,m)$-modules is ...
- 57.6k
8
votes
2
answers
217
views
Flipping Hilbert series of semigroup rings
I'll first give intuition, and then give a precise statement.
For $|z|<1$, we have $\sum_{i \geq 0} z^i = 1/(1-z)$. For $|z|>1$, we have $\sum_{i<0} (-1) z^i=1/(1-z)$. Thus, the two ...
- 156k
5
votes
1
answer
220
views
When are these rings regular?
Let $R$ be a noetherian regular domain. Suppose that $a, b \in R$, with $b \neq 0$, and consider the ring $S:=R[\frac{a}{b}]=R[X]/(bX-a)$. Is $S$ regular? If this is not the case are there some ...
- 5,846
7
votes
3
answers
2k
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Products of Ideal Sheaves and Union of irreducible Subvarieties
Assume I have a nonsingular, irreducible, algebraic variety $X$ and irreducible, nonsingular subvarieties $Z_1,\ldots,Z_k\subseteq X$. Let $\mathcal{I}_i$ be the ideal sheaf of $Z_i$ and $\mathcal{I}:=...
- 39.3k
1
vote
0
answers
138
views
Bases of Ideals With no Monomials
Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in ...
- 345
10
votes
1
answer
1k
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Formally smooth morphisms, the cotangent complex, and an extension of the conormal sequence
I'm reading Daniel Quillen's paper "Homology of commutative rings," in which he proves:
A finitely presented morphism of rings $A \to B$ is
Formally etale iff $L_{B/A}$ (this denotes the cotangent ...
- 19.6k
2
votes
0
answers
245
views
Is simplicity preserved under completion of the base ring?
Let $(A,\mathfrak{m})$ be a noetherian local ring and $R$ be an $A$-algebra, which is finitely generated generated as an $A$-module (module finite $A$-algebra). Let $\widehat{A}$ be the $\mathfrak{m}$-...
- 1,391
37
votes
3
answers
3k
views
What does it mean geometrically that an element in a domain is irreducible?
Consider a domain $A$ and a non-zero element $f\in A$. That element $f$ is prime if and only if the subscheme $V(f)\subset \operatorname{Spec}(A)$ is integral and this is a completely satisfactory ...
- 20.8k
2
votes
1
answer
456
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Generic liftings of a regular sequence on the initial ideal
Hi everyone,
I've got a question about explicitly lifting regular sequences. Let $I$ be an ideal in a polynomial ring $S$ with some term order. We'll denote the initial ideal by $in(I)$. It is ...
- 46
4
votes
1
answer
662
views
Modules with flat duals
Let $R$ be a commutative ring, $M$ an $R$-module, $M^*=Hom_R(M,R)$ its dual. What are sufficient (and possibly necessary) conditions on $M$ that ensure that $M^*$ is flat? Is there a name for such ...
CommunityBot
- 1
3
votes
1
answer
268
views
Universal catenarity and Laurent algebras
A Noetherian (commutative) ring $A$ is called universally catenary if every $A$-algebra of finite type is catenary. If one wants to know whether $A$ is universally catenary, then this definition ...
- 19.6k